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Chapter 6: Problem 16

Use the graphical method to solve the given system of equations for \(x\) and \(y .\) $$\left\\{\begin{array}{c}5 x+y=10 \\ x+y=6\end{array}\right.$

Short Answer

Expert verified
The solution is the point of intersection of the two lines.

Step by step solution

01

- Rewrite Equations in Slope-Intercept Form

First, rewrite each equation in the form of y = mx + b (slope-intercept form). For the first equation: 5x + y = 10, subtract 5x from both sides: y = -5x + 10. For the second equation: x + y = 6, subtract x from both sides: y = -x + 6.
02

- Plot the First Equation

For the equation y = -5x + 10, plot several points. When x = 0, y = 10 (0,10). When x = 2, y = 0 (2,0). Draw a line through these points.
03

- Plot the Second Equation

For the equation y = -x + 6, plot several points. When x = 0, y = 6 (0,6). When x = 6, y = 0 (6,0). Draw a line through these points.
04

- Identify the Intersection

Look for the point where the two lines intersect. This point is the solution to the system of equations.
05

- Verify the Solution

Substitute the coordinates of the intersection point back into the original equations to ensure they satisfy both equations.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

System of Equations
A system of equations consists of two or more equations that have at least one solution in common. The goal is often to find the point(s) where both equations are true at the same time. In the given exercise, we have a system of two linear equations: \[\begin{array}{l}5x + y = 10 \ x + y = 6 \end{array}\]
By solving this system, we find the values of \(x\) and \(y\) that satisfy both equations. There are various methods to solve systems of equations, including substitution, elimination, and the graphical method, which involves plotting the equations on a coordinate plane and finding their intersection point.
Slope-Intercept Form
The slope-intercept form of a linear equation is \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept. The slope tells us how steep the line is, and the y-intercept is where the line crosses the y-axis. For the equations in this exercise:
  • First Equation: \(y = -5x + 10\)
  • Second Equation: \(y = -x + 6\)
By converting the given equations into this form, it becomes easier to plot them on a graph. You simply need to identify the slope and y-intercept for each line and use these to plot points.
Plotting Points
Plotting points is a crucial step in solving equations graphically. For each equation, choose values for \(x\) and calculate the corresponding \(y\) values:
  • For \(y = -5x + 10\), when \(x = 0\), \(y = 10\) (point: (0,10)). When \(x = 2\), \(y = 0\) (point: (2,0)).
  • For \(y = -x + 6\), when \(x = 0\), \(y = 6\) (point: (0,6)). When \(x = 6\), \(y = 0\) (point: (6,0)).
Plot these points on the coordinate plane. Draw a straight line through the points of each equation. The visual representation will help you to easily understand where both lines intersect.
Intersection Point
The intersection point of two lines on a graph represents the solution to the system of equations. To find the intersection, observe where the lines cross. For our example:
  • First Equation: \(y = -5x + 10\)
  • Second Equation: \(y = -x + 6\)
The point where these lines intersect is the solution. In this case, after plotting, you will find they intersect at \((2, 4)\).

It's important to verify this solution by substituting \(x = 2\) and \(y = 4\) back into the original equations to ensure they satisfy both equations:
  • For 5x + y = 10: \(5(2) + 4 = 10\)
  • For x + y = 6: \(2 + 4 = 6\)
Both equations are true, confirming that \((2, 4)\) is the correct solution!

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Most popular questions from this chapter

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. Mrs. Thomas has \(\$ 15,000\) to invest. She will invest part of it in a corporate bond that pays \(8 \%\) interest per year and the rest in a stock that pays \(11 \%\) interest per year. How should she divide up the \(\$ 15,000\) so that her total yearly interest will be \(10 \%\) of her investments?

The U.S. House of Representatives has 435 members, all of whom voted on a certain bill. The ratio of those who voted in favor of the bill to those who voted against it was 8 to \(7 .\) How many were in favor of the bill?

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. A small company has budgeted \(\$ 6,000\) per month to lease vehicles. On this budget, the company can lease 12 cars and 4 trucks each month, or 8 cars and 6 trucks. Find the monthly cost to lease a car and to lease a truck.

In Exercises \(29-62,\) solve the system of equations using any method you choose. $$\left\\{\begin{array}{l} \frac{3 w}{5}+\frac{4 z}{3}=44 \\ \frac{5 w}{8}+\frac{7 z}{6}=45 \end{array}\right.$$

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. A mathematics department has budgeted \(\$ 10,000\) to purchase computers and printers. On this fixed budget they can purchase 10 computers and 10 printers, or they can purchase 12 computers and 2 printers. Find the individual costs of a computer and printer.
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